Thermodynamics à la Souriau on Kähler Non-Compact Symmetric Spaces for Cartan Neural Networks

In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces U/H that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a clear-cut distinction between the generalized thermodynamics associated with Integrable Dynamical Systems and the challenging proposal of Gibbs probability distributions on U/H provided by generalized thermodynamics à la Souriau. Our main result is the proof that U/H.s supporting such Gibbs distributions are only the Kähler ones. Furthermore, for the latter, we solve the problem of determining the space of temperatures, namely, of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of U of a positivity domain in the Cartan subalgebra Cc⊂H of the maximal compact subalgebra H⊂U. We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi-Vesentini manifolds utilizing Paint Group symmetry. Furthermore, we claim that Rao's, Chentsov's, and Amari's Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. In particular, we provide an explicit study of thermodynamical geometry for the Poincaré plane. The key feature of the Gibbs probability distributions in this setup is their covariance under the entire group of symmetries U. The partition function is invariant against U transformations, and the set of its arguments, namely the generalized temperatures, can always be reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group H⊂U.